Igor Pikovski et al have proposed a test for Planck scale physics using recent advances in quantum optics. The framework they use is a modification of quantum mechanics, expressed by a deformation of the canonical commutation relation, that takes into account that the Planck length plays the role of a minimal length. This is one of the most promising routes to quantum gravity phenomenology, and I was excited to read the article.

In their article, the authors claim that their proposed experiment is feasible to "probe the possible effects of quantum gravity in table-top quantum optics experiment" and that it reaches a "hitherto unprecedented sensitivity in measuring Planck-scale deformations." The reason for this increased sensitivity for Planck-scale effects is, according to the authors own words, that "the deformations are enhanced in massive quantum systems."

Unfortunately, this claim is not backed up by the literature the authors refer to.

The underlying reason is that the article fails to address the question of Lorentz-invariance. The deformation used is not invariant under normal Lorentz-transformations. There are two ways to deal with that, either breaking Lorentz-invariance or deforming it. If it is broken, there exists a multitude of very strong constraints that would have to be taken into account and are not mentioned in the article. Presumably then the authors implicitly assume that
Lorentz-symmetry is suitably deformed in order to keep the commutation relations invariant - and in order to test something actually new. This can in fact be done, but comes at a price. Now the momenta transform non-linearly. Consequently, a linear sum of momenta is no longer Lorentz-invariant. In the appendix however, the authors have used the normal sum of momenta to define the center-of-mass momentum. This is inconsistent. To maintain Lorentz-invariance, the modified sum must be used.

This issue cannot be ignored for the following reason. If a suitably Lorentz-invariant sum is used, it contains higher-order terms. The relevance of these terms does indeed increase with the mass. This also means that the modification of the Lorentz-transformations become more relevant with the mass. Since this is a consequence of just summing up momenta, and has nothing in particular to do with the nature of the object that is being studied, the increasing relevance of corrections prevents one from reproducing a macroscopic limit that is in agreement with our
knowledge of Special Relativity. This behavior of the sum, whose use, we recall, is necessary for Lorentz-invariance, is thus highly troublesome. This is known in the literature as the "soccer ball problem." It is not mentioned in the article.

If the soccer-ball problem persists, the theory is in conflict with observation already. While several suggestions have been made how this problem can be addressed in the theory, no agreement has been reached to date. A plausible and useful ad-hoc suggestion that has been made by Magueijo and Smolin is that the relevant mass scale, the Planck mass, for N particles is rescaled to N times the Planck mass. Ie, the scale where effects become large moves
away when the number of particles increases.

Now, that this ad-hoc solution is correct is not clear. What is clear however is that, if the theory makes sense at all, the effect must become less relevant for systems with many constituents. A suppression with the number of constituents is a natural expectation.

If one takes into account that for sums of momenta the relevant scale is not the Planck mass, but N times the Planck mass, the effect the authors consider is suppressed by roughly a factor 1010. This means the existing bounds (for single particles) cannot be significantly improved in this way. This is the expectation that one can have from our best current understanding of the theory.

This is not to say that the experiment should not be done. It is always good to test new parameter regions. And, who knows, all I just said could turn out to be wrong. But it does mean that based on our current knowledge, it is extremely unlikely that anything new is to be found there. And vice versa, if nothing new is found, this cannot be used to rule out a minimal length modification of quantum mechanics.

(This is not the first time btw, that somebody tried to exploit the fact that the deviations get larger with mass by using composite systems, thereby promoting a bug to a feature. In my recent review, I have a subsection dedicated to this.)

You can simply watch the surface of superfluid helium - this surface will appear rough because the space-time is grainy at the 2 mm scale already (due the CMBR noise). This graininess indeed goes to Planck scale - but I don't think, we can prove the existence of this scale with experiments at 10E+33 larger scale directly.

The paper that the article you link to refers to is, I believe, this one. Which, however, has nothing to do with holography. The article you link to has mixed up several different things there. The GRB test is about a specific sort of Lorentz-invariance violation. Hogan claims (but doesn't show) that his holographic noise doesn't violate Lorentz-invariance.

People talk about "space-time graininess" as if that's a technical term, I find this very disturbing. It's just an all-encompassing expression for any Planck-scale effects that come from a dynamical or at least not-smooth background spacetime. There's dozens of different models for this, and you can't rule model A out with a test for model B. Best,

thanks for following up on this paper! I only learned through your blog post that it is now in Nature physics, which leaves me rather confused.

I seem to be credited in the acknowledgments for long and painful discussions in which I tried to discourage the first author (ordering seems important in this community ;)) from taking "modified commutators" such as proposed e.g. by Maggiore seriously as a "prediction" from quantum gravity. Look for instance at hep-th/9309034 which "derives"

[x,p]=iħ(1+E^2/k^2)^(1/2)

where E^2=p^2+m^2 is already rather obviously not a Lorentz invariant quantity. I tried to insist on the obvious problems for applying such commutators to macroscopic systems, as this paper proposed to do.

It now looks rather shocking that Nature physics would publish this - but the Maggiore paper is listed in INSPIRE with 172 citations too!

Seems that there's a lot of work for theorists like yourself to explain to the world that not every ad-hoc modification of the Heisenberg commutation relations is exciting "quantum-gravity phenomenology...."

Thanks for the background... The relation is not Lorentz-invariant under the normal Lorentz-transformation. If you use a suitably modified transformation, it can be Lorentz-invariant. Unfortunately, this is never mentioned in Pikovski et al's paper. You can either use the normal Lorentz-transformation and be fine with macroscopic systems, or use the modified transformation and be fine with Lorentz-invariance, but the combination the authors use is not consistent.

Either way, the literature on the topic is a mess. I happen to know that there's a review in the making (not by me) on the topic, which will hopefully sort this out!

Never mind Lorentz invariance, what about composite systems? I ought to have a modified coproduct structure in order to make the commutation relations consistent -- or, as you say, the relevant scale should become N times the Planck mass, which still leaves me with the conceptually disturbing conclusion that I need to know what is 'elementary' and what isn't.This is the step in that paper that makes no sense at all -- the interpretation of the x's and p's as macroscopic center-of-mass coordinates.

I'm sufficiently annoyed at this to want to write a comment to Nature physics. Not sure if this is such a great idea though...

Well, I did write a comment to the editor... Several weeks ago already. I posted this here because I didn't get a reply and was frustrated.

However, it seems I wasn't patient enough because just yesterday I got a reply from the editor. It seems they took my comment very seriously and did actually send it to a reviewer (which I hadn't expected). The reviewer agreed on the technical details but didn't think that was serious enough "wrongness" to merit correction. (I'm paraphrasing this, but that's the essence.) My intention actually hadn't been to publish a commentary, but mostly to inform whoever was responsible for this publication. As I mentioned in my post, that composite systems "enhance" the Planck scale effects has been "re-discovered" a couple of times, and I'm afraid if one paper got published in Nature, more will follow. So now the editor knows to be careful...

Btw, do you think the issue with composite problems will be solved any time soon? Because that would shed a completely different light on the situation. Best,

I'm not familiar with all of the (vast) literature on this, the papers I have looked at avoid discussing composite systems. It shouldn't be too difficult to cook up a coproduct that allows you to maintain the same modified commutator for composite systems, but I think this just moves the problem to a different place: Deviations of such a coproduct from the standard one should scale with E/(m_Pl c^2), or p/(m_Pl c), or something like that. For macroscopic systems, this would mean violations of momentum conservation growing with the size of the system. Not too compelling...

I read your blog entry on Pikovski et. al. “Probing Planck-scale physics with quantum optics” with great interest. I would like to make the following comments that hopefully clarify the motivation and the analysis behind our paper.

1. It is incorrect that we assume that the centre of mass degree of freedom is given as a linear (or any other) sum of momenta of individual constituents. We do not need to make any such assumption. The centre of mass degree of freedom of the mirror is defined operationally: it is the degree of freedom with which light interacts in the experiment and has been confirmed in numerous experiments thus far. How this degree of freedom arises from the fundamental degrees of freedom of constituting particles is an interesting and as you mentioned still open theoretical question, but not decisive for our argument (we do mention in the Appendix that if the centre of mass degree of freedom is given as a linear sum of momenta of individual constituents – anticipating all the problems that you mention – and the deformation applies to individual constituents, then the predicted effect would be smaller, with the exact amount depending on the underlying theory used.)

2. The proposed experiment can achieve a very high sensitivity, and it can easily compete in the precision of measuring violations of Lorentz invariance that are typically only considered in astronomical observations. But, even if the latter could have better bounds in some ranges of parameters, astronomical observations and other tests of possible consequences of quantum gravity are mostly performed in the domain of validity of classical physics. We think that direct measurements of the non-commutativity of canonical observables on quantum states has its own explanatory power that does not rely on any particular assumed transition from quantum to classical. The latter is not trivial and might be modified in presence of quantum gravity effects.

3. As you mention we do not have a theory with a consistent description of the composite system. The theories of commutator deformations do not precisely specify in which domain they would be applicable, or to which degree of freedom, and it is up to experiments to probe any possibility. It might be that the deformation increases with mass until some maximal mass and then saturates or decreases, such that it is not observable in the classical domain. Honestly, nobody knows what the correct theory would be. Our experiment can test the deformation in the particular range of the experiment which we precisely define in the paper. In this respect it is also important to note that in our setup the predicted effect due to some of the considered deformations of the commutation relation actually decreases with the mass as 1/m. That means that the effect would be larger, the lighter the particle is, provided the interaction with the particle can be realized as required in our paper.

With best regards,

Caslav Brukner (one of the authors of the paper) University of Viennahttp://quantumfoundations.weebly.com

Thanks for your comment. What you say is simply wrong and just reinforces my impression that you don't know very much about the theory you are dealing with. How the collective degrees of freedom arise from the fundamental ones is *the* decisive question to decide whether the experiment you propose will be able to test any interesting range of parameter space, because the models that you refer to always talk about the fundamental degrees of freedom, whereas you don't. The way the collective degrees of freedom scale in your paper is not a consistent and viable theoretical model. What I am saying is that, if you know something about the theory, you already know it is extremely implausible there is anything to be found there.

" It might be that the deformation increases with mass until some maximal mass and then saturates or decreases, such that it is not observable in the classical domain. Honestly, nobody knows what the correct theory would be."

The one thing we know is that the effect can't continue to increase with the mass. That it increases just for some bit and is relevant just in the regime that you want to measure, and then decreases is logically possible but there is not a single model that leads to such a behavior and it is, at least to me, very implausible. You could equally well go and say, there might be a model in which quantum mechanics doesn't work just exactly in that one restaurant in New York City, so we should go and measure it.

For the sake of the interested reader I will comment on the points you raise, and ignore the noncollegial sections in your answer. You mention interpretational issues as if they are facts, but there is no consensus whatsoever as to how to construct a theory with modified commutators. You say that they should be exclusively applied to "fundamental" degrees of freedom, but it is absolutely not clear what these degrees of freedom are and on what basis are chosen as such. Additionally, once one declares some degrees to be “fundamental” the theory runs into a problem of even properly defining compositions of them, not to speak about predicting the physics of them. The form of the modifications is also usually only given to the first order (around the Planck mass where our experiment could provide results), notwithstanding, you say that an initial increase followed by a saturation or decrease for higher orders cannot be. Our attitude is that only empirical observations can back up such claims.

What is "noncollegial" about my answer? If I think something is wrong, please allow me to say it is wrong.

"you say that an initial increase followed by a saturation or decrease for higher orders cannot be. Our attitude is that only empirical observations can back up such claims."

You misunderstood what I said. I did not say "it cannot be," I said it is implausible and I explained you exactly why, based on what we know of the theory you are working with. Look, the situation with the phenomenology of quantum gravity is that you have a model, and the model has parameters, and to see if it is interesting testing the model in some parameter range, you need to know what is the range of the parameters that you expect. I have been explaining why the range of the parameters that you expect in this case is many, many orders of magnitude away from what you can experimentally probe, because the effects of the theory that you are referring to in your paper cannot increase with the number of constituents as you claim they do, that would be incompatible with observations we already have done.

"Additionally, once one declares some degrees to be “fundamental” the theory runs into a problem of even properly defining compositions of them, not to speak about predicting the physics of them. "

Right. What it means in plain words is that there is no theory that you are using. You are writing down some commutation relation that does not follow from anything and does not even have a motivation. What you still don't seem to have understood though is that the addition law that you would need to get the scaling that you are working with is not compatible with other observations, for example that momentum conservation works nicely and linearly as Newton told us in all experiments we have ever done, and that bound states exists to begin with. Please note, the addition law has nothing to do with quantum effects as you seem to assume, which is why exporting it into the supplement and not discussing the issue is a major omission.

Besides that, I also said, explicitly (read my post) this is not to say that the experiment should not be done. Do whatever you like. But if you don't find anything, then don't go around and say you have ruled out a model which you didn't actually use to begin with. Best,